Are you talking about Assateaque Nationa Seashore or the state park. They are right next to each other. To the best of my knowledge there is no restriction at the National Park and the state park just open up more sites for dog owners

You probably won't get in on a weekend however during the week you should find spaces. It is one of my favorites especially in the spring and fall. If you have not already I would recommend that you go online to recreation.gov to see what is a available or call them. There are no hook-ups so you need to bring your own water and generators. There are two dump stations and also two places to refill your water tanks.
Another suggestion is you might want to try Delaware Seashore state park. They just added new sites and I think with water and electricity. It is in Rehobeth on the mouth of the Indian River. The ocean is cross the road from it. Lot of fishing in the area.